DOCSLIB.ORG

Sets of Minimal Hausdorff Dimension for Quasiconformal Maps

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Sets of Minimal Hausdorff Dimension for Quasiconformal Maps PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 128, Number 11, Pages 3361{3367 S 0002-9939(00)05433-2 Article electronically published on May 18, 2000 SETS OF MINIMAL HAUSDORFF DIMENSION FOR QUASICONFORMAL MAPS JEREMY T. TYSON (Communicated by Albert Baernstein II) Abstract. For any 1 ≤ α ≤ n, there is a compact set E ⊂ Rn of (Hausdorff) dimension α whose dimension cannot be lowered by any quasiconformal map f : Rn ! Rn. We conjecture that no such set exists in the case α<1. More generally, we identify a broad class of metric spaces whose Hausdorff dimension is minimal among quasisymmetric images. 1. Introduction Quasiconformal homeomorphisms of a Euclidean space Rn, n ≥ 2, can distort the Hausdorff dimensions of subsets. While the dimensions of sets of Hausdorff dimension zero or n must be preserved, Gehring and V¨ais¨al¨a [5, pp. 505-506] n construct, for any β 2 (0;n), a compact set Eβ ⊂ R with dim Eβ = β and, for 0 n n any β,β 2 (0;n), a quasiconformal map f : R ! R with Eβ0 = fEβ. Recently, Bishop [2] showed that the dimension of any compact set E ⊂ Rn of positive dimension can be raised arbitrarily close to n by a quasiconformal (quasisymmetric if n = 1) homeomorphism of Rn. On the other hand, for certain values of α 2 (0;n), one can construct compact sets E ⊂ Rn of dimension α whose dimension cannot be lowered by any quasicon- formal mapping of the ambient space. For example, consider the closed unit k-cube [0; 1]k, k =1; 2;::: ;n−1. These examples have Hausdorff dimension equal to their topological dimension and thus the Hausdorff dimension cannot be lowered by any homeomorphism of Rn. A different example (arising from the theory of removable sets for quasiconformal maps) was given by Bishop [1] in the case n =3,α =2. Bishop [2] asks whether there exist examples of this phenomenon for all 0 <α<n. The following result is a simple consequence of the main theorem in this paper: Theorem 1.1. Let 1 ≤ α ≤ n. There is a compact set E ⊂ Rn with Hausdorff dimension α so that dim fE ≥ α for all quasiconformal maps f : Rn ! Rn. When α>1 we may take the set E to be the Cartesian product of any Ahlfors (α − 1)-regular set in Rn−1 with the unit interval [0; 1]; for the definition of Ahlfors Received by the editors October 15, 1998 and, in revised form, January 15, 1999. 2000 Mathematics Subject Classification. Primary 30C65; Secondary 28A78. Key words and phrases. Hausdorff dimension, quasiconformal maps, generalized modulus. The results of this paper form part of the author's doctoral dissertation at the University of Michigan. The author was supported by a National Science Foundation Graduate Research Fellowship and a Sloan Doctoral Dissertation Fellowship. c 2000 American Mathematical Society 3361 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3362 JEREMY T. TYSON regularity, see section 2 (for the existence of such sets, note that the set Eβ in the above example of Gehring and V¨ais¨al¨aisβ-regular). We will prove a more general result (Theorem 3.4) using the machinery of the generalized (or discrete) modulus introduced in [9] and developed in [13]. The restriction to subsets of a Euclidean space is not necessary; we give a condition on metric spaces which is sufficient to guarantee that the Hausdorff dimension is minimal among quasisymmetric images (recall that in Rn, n ≥ 2, the classes of quasiconformal and quasisymmetric maps agree). It should be emphasized that Theorem 1.1 is not new; the result appears in works of Pansu [9] and Bourdon [3] (see, e.g., [9, Example 3.4] and [3, Example 1.7(b)]). These authors give a family of conditions (Lemma 3.9) on an arbitrary metric space which imply nontrivial lower bounds on the Hausdorff dimensions of quasisymmetric images. With an additional (weak) assumption, we show in Proposition 3.11 that the hypotheses of Theorem 3.4 are implied by the \best" condition of Bourdon and Pansu. The above-mentioned examples which illustrate Theorem 1.1 satisfy this \best" condition of Bourdon-Pansu. The restriction to dimensions larger than one is necessary in our proof, which relies on the geometric characterization of quasiconformality in terms of curve fam- ilies. We conjecture that no such example exists for dimensions strictly smaller than one, more precisely: Conjecture 1.2. Let n ≥ 1 and >0.IfacompactsetE ⊂ Rn has Hausdorff dimension strictly smaller than one, then there exists a quasiconformal (quasisym- metric if n =1)mapf : Rn ! Rn so that fE has Hausdorff dimension smaller than . Note that such a set must be totally disconnected. For related results, see [12]. 2. Notation and definitions In a metric space X,wedenotebyB(x; r) the closed ball with center x 2 X and radius r>0. For Y ⊂ X,wewritediamY for the diameter of Y .IfB = B(x; r) is a ball in X,wewriteCB for the dilated ball B(x; Cr). For x 2 X and r>0, we have diam B(x; r) ≤ 2r, but no lower bound for diam B(x; r) need hold in general. We call X (c1-)uniformly perfect,0<c1 ≤ 2, if diam B(x; r) ≥ c1r for every x 2 X and 0 <r≤ diam X. Every connected space is 1-uniformly perfect. For α>0, we denote by Hα the Hausdorff α-measure on X and by dim X the Hausdorff dimension of X [4, x2.10.2(1)]. For connected sets E,wehavetherelation diam E ≤H1(E)[4,x2.10.12]. For α>0, we call a Borel measure µ on X (Ahlfors) α-regular if there exists a constant C0 ≥ 1 (called a regularity constant for X)sothat 1 α α (2.1) r ≤ µ(B(x; r)) ≤ C0r C0 for all x 2 X and 0 <r<diam X.Ifµ is a Borel measure on X satisfying (2.1), then Hα also satisfies (2.1) (possibly with a different regularity constant) and µ and Hα are equivalent. Thus α-regular spaces have Hausdorff dimension α. Every α-regular space with regularity constant C0 is c1-uniformly perfect for each −2/α c1 <C0 . License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use SETS OF MINIMAL HAUSDORFF DIMENSION 3363 We call a homeomorphism f : X ! Y between metric spaces (η-)quasisymmetric, where η :[0; 1) ! [0; 1) is a homeomorphism, if (2.2) jx − y|≤t jx − zj)jf(x) − f(y)|≤η(t) jf(x) − f(z)j for any x; y; z 2 X and t>0 [11]. By making η larger if necessary, we can (and will) always assume that η(t) ≥ t for t>0 and that both f and f −1 satisfy (2.2). If X ! 0 is c1-uniformly perfect and f : X Y is η-quasisymmetric, then Y is c1-uniformly 0 0 perfect for some c1 = c1(c1,η) > 0. The Hausdorff dimension of a metric space is not a quasisymmetric invariant. Following Pansu [9, D´efinition 3.1], we define the conformal dimension of a metric space X to be C dim X =infdim Y; Y where the infimum is taken over all spaces Y which are quasisymmetrically equiv- alent to X. For any space X, C dim X ≥ dimT X, where dimT X denotes the topological dimension of X [7]. 3. The main result Let (X; µ) be a metric measure space. For a family Γ of curves in X and p ≥ 1, we denote by Modp Γthep-modulus of Γ [14, x6.1], [6, x2.3]. (Throughout this paper, we restrict ourselves to consider only curves whose parametrization is one-to-one.) Definition 3.1. We say that X has nontrivial p-modulus, p ≥ 1, if Modp Γ > 0for some family of curves Γ in X. Example 3.2. Let α>1. Let Z be any compact (α − 1)-regular metric space and let X be the product space Z ×[0; 1] (endowed with any of its standard bi-Lipschitz equivalent metrics). Then X is a compact, α-regular space and the family of curves Γ=fγz :[0; 1] ! Xjγz(t)=(z;t);z 2 Zg has positive α-modulus. Note that for any n ≥ 2and1<α≤ n,wemaychoose Z ⊂ Rn−1 (see the introduction) and so X ⊂ Rn. When 1 <α<2, the preceding example is not connected. We next give con- nected examples for each 1 <α≤ n. Example 3.3. Let 1 <α≤ n and choose a compact (α − 1)-regular Borel set Z ⊂ Sn−1.Letν be an (α − 1)-regular measure on Z and let X = ftz : z 2 Z; 0 ≤ t ≤ 1}⊂Rn n be the cone on Z. Here, for z =(z1;::: ;zn) 2 R and t ≥ 0, we write tz = (tz1;::: ;tzn). We define a measure µ on X as follows: for Borel sets A ⊂ [0; 1] and B ⊂ Z, define the spherical product set A · B = ftz : t 2 A; z 2 Bg and set Z µ(A · B)=ν(B) · rn−1dr: A The collection of spherical product sets generates the Borel σ-algebra on X and µ is an α-regular Borel measure. Moreover, the family of curves Γ=fγz :[1=2; 1] ! Xjγz(t)=tz; z 2 Zg has positive α-modulus, by an argument similar to [14, Example 7.5]. Theorem 1.1 is a consequence of these examples and the following result.
Load more

Recommended publications
  • Hausdorff Dimension of Singular Vectors
    HAUSDORFF DIMENSION OF SINGULAR VECTORS YITWAH CHEUNG AND NICOLAS CHEVALLIER Abstract. We prove that the set of singular vectors in Rd; d ≥ 2; has Hausdorff dimension d2 d d+1 and that the Hausdorff dimension of the set of "-Dirichlet improvable vectors in R d2 d is roughly d+1 plus a power of " between 2 and d. As a corollary, the set of divergent t t −dt trajectories of the flow by diag(e ; : : : ; e ; e ) acting on SLd+1 R= SLd+1 Z has Hausdorff d codimension d+1 . These results extend the work of the first author in [6]. 1. Introduction Singular vectors were introduced by Khintchine in the twenties (see [14]). Recall that d x = (x1; :::; xd) 2 R is singular if for every " > 0, there exists T0 such that for all T > T0 the system of inequalities " (1.1) max jqxi − pij < and 0 < q < T 1≤i≤d T 1=d admits an integer solution (p; q) 2 Zd × Z. In dimension one, only the rational numbers are singular. The existence of singular vectors that do not lie in a rational subspace was proved by Khintchine for all dimensions ≥ 2. Singular vectors exhibit phenomena that cannot occur in dimension one. For instance, when x 2 Rd is singular, the sequence 0; x; 2x; :::; nx; ::: fills the torus Td = Rd=Zd in such a way that there exists a point y whose distance in the torus to the set f0; x; :::; nxg, times n1=d, goes to infinity when n goes to infinity (see [4], chapter V).
    [Show full text]
  • 23. Dimension Dimension Is Intuitively Obvious but Surprisingly Hard to Define Rigorously and to Work With
    58 RICHARD BORCHERDS 23. Dimension Dimension is intuitively obvious but surprisingly hard to define rigorously and to work with. There are several different concepts of dimension • It was at first assumed that the dimension was the number or parameters something depended on. This fell apart when Cantor showed that there is a bijective map from R ! R2. The Peano curve is a continuous surjective map from R ! R2. • The Lebesgue covering dimension: a space has Lebesgue covering dimension at most n if every open cover has a refinement such that each point is in at most n + 1 sets. This does not work well for the spectrums of rings. Example: dimension 2 (DIAGRAM) no point in more than 3 sets. Not trivial to prove that n-dim space has dimension n. No good for commutative algebra as A1 has infinite Lebesgue covering dimension, as any finite number of non-empty open sets intersect. • The "classical" definition. Definition 23.1. (Brouwer, Menger, Urysohn) A topological space has dimension ≤ n (n ≥ −1) if all points have arbitrarily small neighborhoods with boundary of dimension < n. The empty set is the only space of dimension −1. This definition is mostly used for separable metric spaces. Rather amazingly it also works for the spectra of Noetherian rings, which are about as far as one can get from separable metric spaces. • Definition 23.2. The Krull dimension of a topological space is the supre- mum of the numbers n for which there is a chain Z0 ⊂ Z1 ⊂ ::: ⊂ Zn of n + 1 irreducible subsets. DIAGRAM pt ⊂ curve ⊂ A2 For Noetherian topological spaces the Krull dimension is the same as the Menger definition, but for non-Noetherian spaces it behaves badly.
    [Show full text]
  • A Metric on the Moduli Space of Bodies
    A METRIC ON THE MODULI SPACE OF BODIES HAJIME FUJITA, KAHO OHASHI Abstract. We construct a metric on the moduli space of bodies in Euclidean space. The moduli space is defined as the quotient space with respect to the action of integral affine transformations. This moduli space contains a subspace, the moduli space of Delzant polytopes, which can be identified with the moduli space of sym- plectic toric manifolds. We also discuss related problems. 1. Introduction An n-dimensional Delzant polytope is a convex polytope in Rn which is simple, rational and smooth1 at each vertex. There exists a natu- ral bijective correspondence between the set of n-dimensional Delzant polytopes and the equivariant isomorphism classes of 2n-dimensional symplectic toric manifolds, which is called the Delzant construction [4]. Motivated by this fact Pelayo-Pires-Ratiu-Sabatini studied in [14] 2 the set of n-dimensional Delzant polytopes Dn from the view point of metric geometry. They constructed a metric on Dn by using the n-dimensional volume of the symmetric difference. They also studied the moduli space of Delzant polytopes Den, which is constructed as a quotient space with respect to natural action of integral affine trans- formations. It is known that Den corresponds to the set of equivalence classes of symplectic toric manifolds with respect to weak isomorphisms [10], and they call the moduli space the moduli space of toric manifolds. In [14] they showed that, the metric space D2 is path connected, the moduli space Den is neither complete nor locally compact. Here we use the metric topology on Dn and the quotient topology on Den.
    [Show full text]
  • Lectures on Fractal Geometry and Dynamics
    Lectures on fractal geometry and dynamics Michael Hochman∗ June 27, 2012 Contents 1 Introduction 2 2 Preliminaries 3 3 Dimension 4 3.1 A family of examples: Middle-α Cantor sets . 4 3.2 Minkowski dimension . 5 3.3 Hausdorff dimension . 9 4 Using measures to compute dimension 14 4.1 The mass distribution principle . 14 4.2 Billingsley's lemma . 15 4.3 Frostman's lemma . 18 4.4 Product sets . 22 5 Iterated function systems 25 5.1 The Hausdorff metric . 25 5.2 Iterated function systems . 27 5.3 Self-similar sets . 32 5.4 Self-affine sets . 36 6 Geometry of measures 40 6.1 The Besicovitch covering theorem . 40 6.2 Density and differentiation theorems . 45 6.3 Dimension of a measure at a point . 50 6.4 Upper and lower dimension of measures . 52 ∗Send comments to [email protected] 1 6.5 Hausdorff measures and their densities . 55 7 Projections 59 7.1 Marstrand's projection theorem . 60 7.2 Absolute continuity of projections . 63 7.3 Bernoulli convolutions . 65 7.4 Kenyon's theorem . 71 8 Intersections 74 8.1 Marstrand's slice theorem . 74 8.2 The Kakeya problem . 77 9 Local theory of fractals 79 9.1 Microsets and galleries . 80 9.2 Symbolic setup . 81 9.3 Measures, distributions and measure-valued integration . 82 9.4 Markov chains . 83 9.5 CP-processes . 85 9.6 Dimension and CP-distributions . 88 9.7 Constructing CP-distributions supported on galleries . 90 9.8 Spectrum of an Markov chain .
    [Show full text]
  • On the Computation of the Hausdorff Dimension of the Walrasian Economy:Further Notes
    Munich Personal RePEc Archive On the Computation of the Hausdorff Dimension of the Walrasian Economy:Further Notes Dominique, C-Rene Independent Research 9 August 2009 Online at https://mpra.ub.uni-muenchen.de/16723/ MPRA Paper No. 16723, posted 10 Aug 2009 10:42 UTC ON THE COMPUTATION OF THE HAUSDORFF DIMENSION OF THE WALRASIAN ECONOMY: FURTHER NOTES C-René Dominique* ABSTRACT: In a recent paper, Dominique (2009) argues that for a Walrasian economy with m consumers and n goods, the equilibrium set of prices becomes a fractal attractor due to continuous destructions and creations of excess demands. The paper also posits that the Hausdorff dimension of the attractor is d = ln (n) / ln (n-1) if there are n copies of sizes (1/(n-1)), but that assumption does not hold. This note revisits the problem, demonstrates that the Walrasian economy is indeed self-similar and recomputes the Hausdorff dimensions of both the attractor and that of a time series of a given market. KEYWORDS: Fractal Attractor, Contractive Mappings, Self-similarity, Hausdorff Dimensions of the Walrasian Economy, the Hausdorff dimension of a time series of a given market. INTRODUCTION Dominique (2009) has shown that the equilibrium price vector of a Walrasian pure exchange economy is a closed invariant set E* Rn-1 (where R is the set of real numbers and n-1 are the number of independent prices) rather than the conventionally assumed stationary fixed point. And that the Hausdorff dimension of the attractor lies between one and two if n self-similar copies of the economy can be made.
    [Show full text]
  • Typical Behavior of Lower Scaled Oscillation
    TYPICAL BEHAVIOR OF LOWER SCALED OSCILLATION ONDREJˇ ZINDULKA Abstract. For a mapping f : X → Y between metric spaces the function lipf : X → [0, ∞] defined by diam f(B(x,r)) lipf(x) = liminf r→0 r is termed the lower scaled oscillation or little lip function. We prove that, given any positive integer d and a locally compact set Ω ⊆ Rd with a nonempty interior, for a typical continuous function f : Ω → R the set {x ∈ Ω : lipf(x) > 0} has both Hausdorff and lower packing dimensions exactly d − 1, while the set {x ∈ Ω : lipf(x) = ∞} has non-σ-finite (d−1)- dimensional Hausdorff measure. This sharp result roofs previous results of Balogh and Cs¨ornyei [3], Hanson [11] and Buczolich, Hanson, Rmoutil and Z¨urcher [5]. It follows, e.g., that a graph of a typical function f ∈ C(Ω) is microscopic, and for a typical function f : [0, 1] → [0, 1] there are sets A, B ⊆ [0, 1] of lower packing and Hausdorff dimension zero, respectively, such that the graph of f is contained in the set A × [0, 1] ∪ [0, 1] × B. 1. Introduction For a mapping f : X Y between metric spaces the function lipf : X [0, ] defined by → → ∞ diam f(B(x, r)) (1) lipf(x) = lim inf r→0 r is termed the lower scaled oscillation or little lip function. (B(x, r) denotes the closed ball of radius r centered at x.) Let us mention that some authors, cf. [3], define the scaled oscillations from the version of ωf (cf.(6)) given by ωf (x, r) = sup f(y) f(x) that may be more suitable for, e.g., differentiation con- y∈B(x,r)| − | siderations.
    [Show full text]
  • CHARACTERIZATIONS of SPACES of FINITE HAUSDORFF MEASURE: ONE-DIMENSIONAL CASE Supervisor
    CHARACTERIZATIONS OF SPACES OF FINITE HAUSDORFF MEASURE: ONE-DIMENSIONAL CASE JORDAN DOUCETTE Supervisor: Dr. Murat Tuncali MAJOR RESEARCH PAPER Submitted to the School of Graduate Studies in partial fulfillment of the requirements for the degree of Master of Science in Mathematics Department of Computer Science and Mathematics Nipissing University North Bay, Ontario c Jordan Doucette December 2012 ! Abstract. An intriguing problem that was proposed by Samuel Eilenberg and O.G. Harrold in 1941 involves the characterization of separable metric spaces, whose topological dimension is n, for which there is a metrization so that the n- dimensional Hausdorff measure is finite. The focus of this paper will be to examine several conditions in order to characterize metric continua of finite one-dimensional Hausdorff measure. Our main theorem will consist of showing that for a metric continuum being of finite one-dimensional Hausdorff measure is equivalent to being totally regular, as well as being an inverse limit of graphs with monotone bonding maps. iv ACKNOWLEDGEMENTS First and foremost, I am grateful to my supervisor Dr. Murat Tuncali, for his invaluable support, instruction and encouragement throughout the preparation of this paper. Without Murat’s continual guidance this paper may never have come to fruition. Thanks are also due to my second reader, Dr. Alexandre Karassev, external examiner, Dr. Ed Tymchatyn and examination committee chair, Dr. Vesko Valov for their helpful feedback. I would also like to extend thanks the entire mathematics department at Nipissing University and visiting professors throughout the years, whose teaching has lead me to this point. Lastly, thank you to my family and friends for their reassurance and unwavering confidence in me.
    [Show full text]
  • On the Hausdorff Volume in Sub-Riemannian Geometry
    On the Hausdorff volume in sub-Riemannian geometry Andrei Agrachev SISSA, Trieste, Italy and MIAN, Moscow, Russia - [email protected] Davide Barilari SISSA, Trieste, Italy - [email protected] Ugo Boscain CNRS, CMAP Ecole Polytechnique, Paris, France - [email protected] October 29, 2018 Abstract For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spher- ical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of the unit ball in the nilpotent approximation and it is always a continuous function. We then prove that up to dimension 4 it is smooth, while starting from dimension 5, in corank 1 case, it is 3 (and 4 on every smooth curve) but in general not 5. These results answer to a question addressedC byC Montgomery about the relation between twoC intrinsic volumes that can be defined in a sub-Riemannian manifold, namely the Popp and the Hausdorff volume. If the nilpotent approximation depends on the point (that may happen starting from dimension 5), then they are not proportional, in general. 1 Introduction In this paper, by a sub-Riemannian manifold we mean a triple S = (M, ∆, g), where M is a connected orientable smooth manifold of dimension n, ∆ is a smooth vector distribution of constant rank k<n, satisfying the H¨ormander condition and g is an Euclidean structure on ∆. A sub-Riemannian manifold has a natural structure of metric space, where the distance is the so called Carnot-Caratheodory distance T d(q0,q1) = inf gγ(t)(γ ˙ (t), γ˙ (t)) dt γ : [0, T ] M is a Lipschitz curve, { 0 | → arXiv:1005.0540v3 [math.DG] 15 Apr 2011 Z q γ(0) = q , γ(T )= q , γ˙ (t) ∆ a.e.
    [Show full text]
  • Convex Bodies with Almost All K-Dimensional Sections Polytopes by LEONI DALLA ASH D
    Math. Proc. Camb. Phil. Soc. (1980), 88, 395 395 Printed in Great Britain Convex bodies with almost all k-dimensional sections polytopes BY LEONI DALLA ASH D. G. LARMAN University College, London (Received 12 September 1979) 1. It is a well-known result of V. L. Klee (2) that if a convex body K in En has all its fc-dimensional sections as polytopes (k > 2) then K is a polytope. In (6,7) E. Schneider has asked if any properties of a polytope are retained by K if the above assumptions are weakened to hold only for almost all ^-dimensional sections of K. In particular, for 2-dimensional sections in E3 he asked whether ext K is countable, or, if not, is the 1-dimensional Hausdorff measure of ext K zero? Here we shall give an example (Theorem 1) of a convex body K in E3, almost all of whose 2-sections are polygons but ext K has Hausdorff dimension 1. Our example does not, however, yield a convex body K with ext K of positive 1-dimensional Hausdorff measure although we believe that such a body exists. Using deep measure-theoretic results of J. M. MarstrandO), as extended by P. Mattila(5), we shall prove a very general result (Theorem 2) showing that a convex body K in En has ext (K n L) of dimension at most s for almost all of its ^-dimensional sections L, if, and only if, the dimension of the (n — &)-skeleton of K is at most n — k + s. 3 THEOREM 1.
    [Show full text]
  • A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals
    mathematics Article A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals Mohsen Soltanifar 1,2 1 Biostatistics Division, Dalla Lana School of Public Health, University of Toronto, 620-155 College Street, Toronto, ON M5T 3M7, Canada; [email protected] 2 Real World Analytics, Cytel Canada Health Inc., 802-777 West Broadway, Vancouver, BC V5Z 1J5, Canada Abstract: How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of aleph-two of virtual fractals with a Hausdorff dimension of a bi-variate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions. Keywords: cantor set; fractals; Hausdorff dimension; continuum; aleph-two MSC: 28A80; 03E10 1. Introduction Benoit Mandelbrot (1924–2010) coined the term fractal and its dimension in his 1975 essay on the quest to present a mathematical model for self-similar, fractured ge- ometric shapes in nature [1]. Both nature and the virtual world have prominent examples Citation: Soltanifar, M. A of fractals: some natural examples include snowflakes, clouds and mountain ranges, while Generalization of the Hausdorff some virtual instances are the middle third Cantor set, the Sierpinski triangle and the Dimension Theorem for Deterministic Vicsek fractal. A key element in the definition of the term fractal is its dimension usually Fractals. Mathematics 2021, 9, 1546. indexed by the Hausdorff dimension. This dimension has played vital roles in develop- https://doi.org/10.3390/math9131546 ing the concept of fractal dimension and the definition of the term fractal and has had extensive applications [2–5].
    [Show full text]
  • The Hausdorff Dimension Distribution of Finite Measures in Euclidean Space
    Can. J. Math., Vol. XXXVIII, No. 6, 1986, pp. 1459-1484 THE HAUSDORFF DIMENSION DISTRIBUTION OF FINITE MEASURES IN EUCLIDEAN SPACE COLLEEN D. CUTLER 1. Introduction. Let E be a Borel set of R^. The a-outer Hausdorff measure of E has been defined to be H"(E) = lirn H%{E) where H%E)= inf 2 (</(*,•))" UB^E and each Bt is a closed ball. d(Bt) denotes the diameter of Bt. It is easily seen that the same value Ha(E) is obtained if we consider coverings of E by open balls or by balls which may be either open or closed. By dim(£') we will mean the usual Hausdorff-Besicovitch dimension of E, where dim(E) = sup{a\Ha(E) = oo} = inf{a\Ha(E) = 0}. The following (see [7] ) are well known elementary properties of dim(E): (1) 0 S dim(E) ^ N. (2) if E is countable then dim(E) = 0 while if X(E) > 0 then dim(E) = N (where X denotes iV-dimensional Lebesgue measure, a notation to be maintained throughout this paper). (3) dim^S En) = sup dim(£J. These properties will be used freely without comment in the following. Let ju be a probability measure on R^. In this paper we introduce the notion of the dimension distribution jx of fi. ji is a probability measure on [0, N] and the quantity jl(E) can be interpreted as the proportion of mass of JU supported strictly on sets with dimension lying in E. Associated with JU and JU is a real-valued random variable à (which we will call the dimension concentration map determined by n) and a family [\p(-, a) }, 0 ^ a ^ N, of probability measures on R^ to be referred to as the Received February 19, 1985 and in revised form July 21, 1985.
    [Show full text]
  • What Is Dimension?
    What is dimension? Somendra M Bhattacharjee1;2 1 Institute of Physics, Bhubaneswar 751005, India 2 Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400085, India email:[email protected] This chapter explores the notion of \dimension" of a set. Various power laws by which an Euclidean space can be characterized are used to define dimensions, which then explore different aspects of the set. Also discussed are the generalization to multifractals, and discrete and continuous scale invariance with the emergence of complex dimensions. The idea of renormalization group flow equations can be introduced in this framework, to show how the power laws determined by dimensional analysis (engineering dimensions) get modified by extra anomalous dimensions. As an example of the RG flow equation, the scaling of conductance by disorder in the context of localization is used. A few technicalities, including the connection between entropy and fractal dimension, can be found in the appendices. Contents 5 Dimensions related to physical problems 12 5.1 Spectral dimension . 12 1 Introduction 2 5.1.1 Diffusion, random walk . 12 5.1.2 Density of states . 13 2 Does \dimension" matter? 2 6 Which d? 13 3 Euclidean and topological dimensions 3 6.1 Thermodynamic equation of state . 13 3.1 Euclidean dimension . 3 6.2 Phase transitions . 14 3.1.1 d via analytic continuation . 4 6.2.1 Discrete case: Ising model . 14 3.2 Topological dimension . 4 6.2.2 Continuous case: crystal, xy, Heisen- berg . 15 4 Fractal dimension: Hausdorff, Minkowski (box) dimensions 6 6.3 Bound states in quantum mechanics .
    [Show full text]